OpenAI says an internal model produced a proof for the Navier–Stokes problem

What OpenAI announced

On September 8th, OpenAI said it is sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. According to the company, the proof was produced by an internal OpenAI system and shows that the dynamics of the equations of fluid motion can develop a singularity in finite time. Both a writeup of the proof and a formalization in Lean have been published.

According to the post, the question of whether smooth three-dimensional fluid motion can break down had remained unresolved for roughly 90 years. OpenAI says it used an internal model significantly more capable than GPT-6 Astra, and that it considers it important to inform the world about the pace of AI progress and what to expect from upcoming models.

The problem itself

The Navier–Stokes equations use Newton's second law of motion to describe how fluids move, and treat a fluid as a continuous medium rather than tracking individual molecules. They are used in aircraft design, weather forecasting and the study of blood flow. The open question was whether the equations for a three-dimensional incompressible fluid of constant density can develop a singularity even when the motion starts smoothly.

A singularity means that speeds in the fluid grow without bound within a finite amount of time — and this would have to happen despite viscosity, which tends to smooth out motion. Because a real fluid cannot move infinitely fast, that would mark a breakdown in how the equations model the fluid; to continue modelling the system, one would have to track the behaviour of each particle individually. The equations date to the nineteenth-century work of Claude-Louis Navier and George Gabriel Stokes; in 1934 Jean Leray proved that solutions exist in a generalized sense, and in 2000 the Clay Mathematics Institute named the problem one of seven Millennium Prize Problems.

The result

OpenAI says its system produced an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in a finite time. The fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity. The post presents this as resolving the Navier–Stokes Millennium Prize problem.

The company links the work to its stated goal of empowering scientists to advance research and technology that benefits all of humanity. The writeup of the proof and the formalization are publicly available.